let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds
M1 + M2 is Idempotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds
M1 + M2 is Idempotent
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is Idempotent & M2 is Idempotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) implies M1 + M2 is Idempotent )
assume that
A1: n > 0 and
A2: ( M1 is Idempotent & M2 is Idempotent ) and
A3: M1 commutes_with M2 and
A4: M1 * M2 = 0. (K,n) ; :: thesis: M1 + M2 is Idempotent
A5: M1 * M2 = 0. (K,n,n) by A4;
A6: ( M1 * M1 = M1 & M2 * M2 = M2 ) by A2, Def1;
(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. (K,n))) + (0. (K,n))) + (M2 * M2) by A1, A3, A4, MATRIX_6:35
.= ((M1 * M1) + (0. (K,n))) + (M2 * M2) by A5, MATRIX_3:4
.= M1 + M2 by A6, A5, MATRIX_3:4 ;
hence M1 + M2 is Idempotent by Def1; :: thesis: verum